2.23 Numerical Examples
51
A =
7.3 10
10 7.3
= 53.29 − 100 = −46.71
B = −
20 10
10 7.3
= −(146 − 100) = −46
C =
20 7.3
10 10
= (200 − 73) = 127
A 2 + B 2 + C 2 =
(−46.71)
2
+ (46) 2 + (127) 2 = 142.92
∴ l 3 =
A
√
A 2 + B 2 + C 2
=
−46.71
142.92
= −0.326
m 3 =
B
√
A 2 + B 2 + C 2
=
−46
142.92
= −0.322
n 3 =
C
√
A 2 + B 2 + C 2
=
127
142.92
= 0.888
2.24 Exercises
1. Define stress at a point in a body under the action of external forces.
2. Derive the differential equation of equilibrium in two dimensions.
3. Explain (a) invariants of stress (b) octahedral stresses.
4. What is meant by octahedral shear stress. Arrive at its value in terms of principal
stress.
5. Given the following stress matrix (in kN/m
2 ), obtain the principal stresses and
their direction cosines.
⎡
⎣
10 20 −40
20 −20 −20
−40 −20 10
⎤
⎦
6. Explain spherical and deviatoric stress tensor components.
7. If the stress field is given by
σ x =
w
10I
5x
2
+ 2c
2
y −
w
3I
y
3
σ y = −
w
6I
2c
3
+ 3c
2 y − y
3
τ xy =
w
2I
x
c
2
− y
2
, find the body forces required to satisfy the equilibrium
conditions.